Stability features of inhomogeneous steady-state potential distributions in diode with counter-streaming electron and ion flows

This paper continues studying stability features of steady states of a diode with counter-streaming electron and ion flows. In our recent paper, an integral-differential equation for the potential perturbation amplitude in the mode without potential barriers reflecting charged particles within the plasma was derived. Its exact solution was found for homogeneous steady-state field distribution. In this paper, we propose a semi-analytical method to solve the integral-differential equation for potential perturbation amplitude in the case of inhomogeneous steady-state solutions. It is based on the use of the piecewise linear approximation of the integral operator kernel and the variable coefficient as well as the potential perturbation distribution. A dispersion equation is obtained and five first dispersion branches are constructed. As a result, we have proved that all steady state potential distributions with the values of dimensionless inter-electrode gap up to 10π/√2 are unstable. Numerical calculations of the potential perturbation development confirm analytical results.